The character tables for the symmetry groups of achiral single-walled carbon nanotubes (SWCNT) were studied. It was found that the SWCNT denoted by (n,m) was constructed by rolling the graphene sheet such that the head of the chiral vector coincided with the tail. The achiral SWCNT have a vertical n-fold rotational axis and n horizontal twofold rotational axes. The results show that the character table for the achiral SWCNT with even n encompasses the tables given in the literature.